The 54-year-old, who is the daughter of the late TV legend Magnus Magnusson, criticised the government's response to the near-collapse of the North Atlantic country's banking system, saying that the UK had imposed "crippling" interest rates on the money owed.
But the mother of five is bound by BBC guidelines prohibiting staff from becoming involved in political rows.
Hundreds of thousands of savers in the UK lost money and had their accounts frozen in October 2008 following the collapse of Icesave, the online bank that went bust.
Almost 50 local authorities in this country invested hundreds of millions of pounds in Iceland, some of it the pensions of their employees.
Ms Magnusson, a presenter on Reporting Scotland and Songs of Praise, aired her outspoken views in a letter to a Sunday newspaper.
She wrote: "As a half-Icelander who always loved the affection, respect and mutual interest between my two countries, this is a painful time.
"But can I make it clear that the current argument convulsing Iceland is mainly about how to pay Britain the Icesave money back.
"Nearly all Icelanders entirely accept their obligation.
"But the terms are crippling for a tiny nation. Why does Britain demand an unchangeable interest rate of 5.5 per cent? Why not 3 per cent or 2 per cent?
"We would barely notice the difference, but Iceland certainly would."
In 2008, Ms Magnusson revealed she had invested money left to her by her late father in Icesave. She later recovered the cash.
A spokesman for BBC Scotland confirmed that Ms Magnusson had breached guidelines which prevent editorial staff from engaging in political debate.
He said she had written to the newspaper because she has "written extensively" about Iceland.
But the spokesman added: "Part of her letter relates to public policy which, under the guidelines, is a subject which journalists should not express their opinions about.
"Sally has been asked to refer to the relevant section of guidelines when writing about specialist areas of interest in the future."
The British and Dutch governments agreed last year to compensate savers for the full amount of their losses and have since put pressure on Iceland to repay the debt but, last week, Iceland's president Olafur Ragnar Grimsson blocked an unpopular but previously approved 3.4 billion bank compensation plan for foreign investors.
Mr Grimsson has come under pressure from Icelandic citizens – 60 per cent of voters oppose the plan, which will cost them 11,000 per person.
Frantic negotiations between the governments of Iceland, the Netherlands and Britain have been ongoing since the weekend in the hopes of avoiding a referendum on the issue.
Ms Magnusson declined to comment.